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Sum of n-th powers of divisors of 32.
4

%I #41 Sep 08 2022 08:44:52

%S 6,63,1365,37449,1118481,34636833,1090785345,34630287489,

%T 1103823438081,35253226045953,1127000493261825,36046397799139329,

%U 1153203048319815681,36897992296869404673,1180663682709764194305

%N Sum of n-th powers of divisors of 32.

%H T. D. Noe, <a href="/A034665/b034665.txt">Table of n, a(n) for n = 0..200</a>

%H Quynh Nguyen, Jean Pedersen, and Hien T. Vu, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Pedersen/pedersen2.html">New Integer Sequences Arising From 3-Period Folding Numbers</a>, Vol. 19 (2016), Article 16.3.1. See Table 1.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (63,-1302,11160,-41664,64512,-32768).

%F G.f.: -3*(21504*x^5-27776*x^4+11160*x^3-1736*x^2+105*x-2) / ((x-1)*(2*x-1)*(4*x-1)*(8*x-1)*(16*x-1)*(32*x-1)). - _Colin Barker_, Apr 20 2014

%F a(n) = (2^(6*n) - 1)/( 2^n - 1). Exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 63*x + 2667*x^2 + 97155*x^3 + ... is the o.g.f. for the 5th subdiagonal of triangle A022166, essentially A006110. - _Peter Bala_, Apr 07 2015

%F a(n) = 1 + 2^n + 4^n + 8^n + 16^n + 32^n for n>=0. - _Karl V. Keller, Jr._, Feb 02 2021

%t Total[#^Range[0, 15]&/@Divisors[32]] (* _Vincenzo Librandi_, Apr 17 2014 *)

%t LinearRecurrence[{63,-1302,11160,-41664,64512,-32768},{6,63,1365,37449,1118481,34636833},20] (* _Harvey P. Dale_, Jan 10 2015 *)

%o (Sage) [sigma(32,n)for n in range(0,15)] # _Zerinvary Lajos_, Jun 04 2009

%o (Magma) [DivisorSigma(n,32): n in [0..15]]; // _Vincenzo Librandi_, Apr 17 2014

%o (PARI) a(n)=(64^n-1)/(2^n-1) \\ _Charles R Greathouse IV_, Oct 07 2015

%o (Python) print([1+2**n+4**n+8**n+16**n+32**n for n in range(15)]) # _Karl V. Keller, Jr._, Feb 02 2021

%Y Cf. A006110, A022166.

%K nonn,easy

%O 0,1

%A _N. J. A. Sloane_